Application of Probability

1. Management
at the local Randy Rooster Chicken Hut noted 50 percent of the customers order
the crispy coating and the remainder order original coating.

a. What
kind of probability distribution will you use to solve this problem?

b. What
is the probability that none of the next four customers will request crispy?

c. What
is the probability that at least two of the next four customers will request
crispy?

d. What
is the probability that no more than two will request crispy?

e. Construct
the probability distribution function table and graph the distribution with a
column chart. Comment on the shape of the graph.

2. Fifty
faculty travel expense vouchers were submitted to the financial office during
the summer. Of these 20 contained errors. If five vouchers are selected at
random what is the probability that:

a. None
contain errors.

b. One
contains an error

c. Three
or more contain errors

d. What
kind of probability distribution did you use to solve this problem?

e. Make
a graph of the PDF

3. Textbook
authors and editors work very hard to minimize the number of errors in a text.
However, they are not perfect and it is estimated that there are an average of
0.8 errors per chapter in a typical text book.

a. What
kind of probability distribution will you use to solve this problem?

b. What
is the probability that there are two or less errors in a particular textbook
chapter?

4. Lunch
customers arrive at Noodles and You at an average rate of 2.8 per minute. For
each calculate the probability:

a. What
kind of probability distribution will you use?

b. What
is the probability of exactly five customers arriving in a minute?

c. What
is the probability that no more than five customers will arrive in a given
minute?

d. Create
the graph of the PDF.

5. Bud
and Lou's Construction build garages. They find is takes an average of 32 hours
to build a garage with a standard deviation of two hours. Assuming that the
time to construct follows a normal distribution:

a. What
percent of the garages take from 32 to 34 hours to erect?

b. What
percent of the garages take from 29 to 34 hours to erect?

c. What
percent of the garages take 28.7 hours or less to erect?

d. Of
all the garages, five (5) percent take how many hours or more to erect?

6. The
annual commissions earned by the sales staff of SaleCo follows a normal
distribution. The mean yearly commission is $40,000 with a standard deviation of
$5000.

a. What
percent of the sales reps earn more than $42,000 per year?

b. What
percent of the sales reps earn between $32,000 and $42,000 per year?

c. What
percent of the sales reps earn between $32,000 and $35,000 per year?

d. The
sales manager wants to award the top 20 percent of the sales reps a bonus. What
is the cutoff point for awarding those bonuses?

7. The
fracture strength of a certain type of glass is normally distributed with a
mean of 579 MPa and a standard deviation of 14 MPA:

a. What
is the probability that a randomly chosen piece of glass will break at less
than 579 BPa?

b. More
than 590 BPa?

c. Less
than 600 BPA?

d. Between
550 and 600 BPA?

8. In
a quality test a sample of 100 nuts from a can of mixed nuts found that 19 were
almonds.

a. Construct
a 90 percent confidence interval for the true proportion of almonds.

b. May
normality be assumed? Explain.

c. Using
the proper formula for minimum sample size for a proportion, what sample size
would be needed for a 90 percent confidence interval and an error of +/- .03?

d. Why
does a quality control manager need to understand sampling?

9. You
are asked to determine the typical number of sales of a certain product at the
store you work for. A sample of 50 days over the last quarter shows an average
of 55 units sold per day with a standard deviation of 10 units.

a. Construct
a 95 percent confidence interval around the mean.

b. Construct
a 99 percent confidence interval around the mean.

c. Show
the two limits on a graph of some sort

d. Interpret
your results.

10. Assume instead you selected a
sample of 90 days sales. And that the mean and standard deviation were the same
at 55 and 10 respectively.

a. Calculate
the new 95 and 99 percent confidence intervals.

b. Show
these on a graph of some sort.

c. Explain
why the new intervals are narrower than the previous ones calculated.

11. A recent survey showed 4.6 percent
of the sample of 250 had suffered some kind of identity theft in the past 12
months.

a. Construct
a 90 percent confidence interval for the true proportion of those suffering
identity theft.

b. Can
we assume the distribution of the population is normal? Explain your answer.

12. A study showed the 14 of 180
publically traded business services companies failed a test for compliance with
Sarbanes-Oxley requirements for financial records and fraud protection.
Assuming these are a good random sample of all publically traded companies:

a. Construct
a 90 percent confidence interval for the overall non-compliance proportion.

b. A
95 percent confidence interval.

c. A
99 percent confidence interval.

d. Explain
why these differ.

13. The formula to calculate a minimum
sample size for numerical data is as follows:

Where n is the sample size, z is
the z value for the level of confidence chosen, s is the estimated standard
deviation and E is the allowable error.

a. Using
this formula calculate the minimum sample size for a study when the level of
confidence is 95 percent, the standard deviation is $1000 and the allowable
error is $100. What actual sample size might you suggest? Explain your answer.

b. How
large of a sample size would be needed for a 99 percent level of confidence?
What actual sample size might you suggest? Why is this sample size larger?
Explain your answer.